High Frequency Limit of the Helmholtz Equations

Jean-David Benamou 1 François Castella Thodoros Katsaounis Benoît Perthame 2
1 ONDES - Modeling, analysis and simulation of wave propagation phenomena
Inria Paris-Rocquencourt, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR2706
Abstract : We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method is based on the Wigner Transform, which is a classical tool for evolution dispersive equations. We extend its use to the stationary case after an appropriate scaling of the Helmholtz equation. Several specific difficulties arise here; first, the identification of the source term (which does not share the quadratic aspect) in the limit, then, the lack of L2 bounds which can be handled with homogeneous Morrey-Campanato estimates, and finally the problem of uniqueness which, at several stage of the proof, is related to outgoing conditions at infinity.
Type de document :
Rapport
[Research Report] RR-3785, INRIA. 1999
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https://hal.inria.fr/inria-00072875
Contributeur : Rapport de Recherche Inria <>
Soumis le : mercredi 24 mai 2006 - 11:09:31
Dernière modification le : jeudi 11 janvier 2018 - 06:20:06
Document(s) archivé(s) le : dimanche 4 avril 2010 - 23:25:56

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Jean-David Benamou, François Castella, Thodoros Katsaounis, Benoît Perthame. High Frequency Limit of the Helmholtz Equations. [Research Report] RR-3785, INRIA. 1999. 〈inria-00072875〉

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